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Smoothness in algebraic geography. (English) Zbl 1032.16012
The most naive way to understand a finite-dimensional associative algebra is to find a basis and analyze its multiplication table. In the modern incarnation, one considers the scheme of all associative \(n\)-dimensional algebras over the field \(k\) as a subscheme of affine space \(\operatorname{Hom}_k(k^n\otimes k^n,k^n)\). Then \(\text{GL}_n(k)\) acts on the \(k\)-rational points of the scheme so that orbits can be interpreted as isomorphism classes of \(n\)-dimensional algebras. So the isomorphism classes of algebras can be understood as a subvariety of \(\operatorname{Hom}_k(k^n\otimes k^n,k^n)\), which is denoted by \(\text{Alg}_r\). The paper analyses the smoothness of some orbits. The particular case where \(X_n\) is the component of the matrix ring is of importance. The paper answers negatively a question of Seshadri and shows that \(X_n\) is singular for \(n\geq 3\). The name Algebraic Geography was given by Flanigan which referred to the study of the variety \(\text{Alg}_r\) in this fashion.

16G10 Representations of associative Artinian rings
16R30 Trace rings and invariant theory (associative rings and algebras)
16P10 Finite rings and finite-dimensional associative algebras
16S80 Deformations of associative rings
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